Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지

THE CONVERGENCE OF HOMOTOPY METHODS FOR NONLINEAR KLEIN-GORDON EQUATION

  • Received : 2009.11.17
  • Accepted : 2009.12.17
  • Published : 2010.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, a Klein-Gordon equation is solved by using the homotopy analysis method (HAM), homotopy perturbation method (HPM) and modified homotopy perturbation method (MHPM). The approximation solution of this equation is calculated in the form of series which its components are computed easily. The uniqueness of the solution and the convergence of the proposed methods are proved. The accuracy of these methods are compared by solving an example.

Keywords

References

  1. S.Abbasbandy ,Modified homotopy perturbation method for nonlinear equations and comparsion with Adomian decomposition method, Appl.Math.Comput. 172 (2006) 431-438. https://doi.org/10.1016/j.amc.2005.02.015
  2. G.Ben-Yu , L.Xun,L. Vazquez , A Legendre spectral method for solving the nonlinear Klein-Gordon equation, Appl.Math.Comput. 15(1996) 19-36.
  3. J.Biazar, H.Ghazvini, Convergence of the homotopy perturbation method for partial differential equations, Nonlinear Analysis: Real World Application. 10(2009) 2633-2640. https://doi.org/10.1016/j.nonrwa.2008.07.002
  4. P.J.Caudrey , I.C.Eilbeck ,J.D. Gibbon, The sine-Gordon equations as a model classical field theory, Nuovo Cimento. 25(1975) 497-511. https://doi.org/10.1007/BF02724733
  5. M.S.H.Chowdhury , I.Hashim , Application of homotopy perturbation method to Klein-Gordon and sine-Gordon equations, Chaos, Solitons and Fractals.39(2009) 1928-1935. https://doi.org/10.1016/j.chaos.2007.06.091
  6. R.K.Dodd , I.C.Eilbeck ,J.D. Gibbon ,H.C. Morris,Solitons and nonlinear wave equations. London: Academoc; 1982.
  7. A.Golbabai , B.Keramati ,Solution of non-linear Fredholm integral equations of the first kind using modified homotopy perturbation method, Chaos Solitons and Fractals. 5 (2009) 2316-2321.
  8. M.Ghasemi ,M.Tavasoli , E.Babolian,Application of He's homotopy perturbation method of nonlinear integro-differential equation, Appl.Math.Comput. 188(2007) 538-548. https://doi.org/10.1016/j.amc.2006.10.016
  9. M.S.Ismail,T. Sarie,Spline difference method for solving Klein-Gordon equations Dirasat, Appl.Math.Comput. 14(1989) 189-196.
  10. M.Javidi , Modified homotopy perturbation method for solving linear Fredholm integral equations, Chaos Solitons and Fractals. 50(2009) 159-165.
  11. S.J.Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman and Hall/CRC Press,Boca Raton. 2003.
  12. S.J.Liao, Notes on the homotopy analysis method:some definitions and theorems, Communication in Nonlinear Science and Numerical Simulation .14(2009)983-997. https://doi.org/10.1016/j.cnsns.2008.04.013
  13. L.Vu.Quoc, S.Li , Invariant-conserving finite difference algorithms for the nonlinear Klein-Gordon equation, Comput.Methods.Appl.Mech.Engrg 107(1993) 314-391.
  14. M.El-Sayed.Salah , The decomposition method for studing the Klein-Gordon equation, Chaos Solitons and Fractals. 18 (2003) 1025-1030. https://doi.org/10.1016/S0960-0779(02)00647-1
  15. A.M.Wazwaz , A first course in integral equations, WSPC, New Jersey; 1997.